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Algebraic decoding of the (41,21,9) quadratic residue code

INFORMATION SCIENCES 2009年第19期179卷 3451-3459页

作者： Lin, Tsung-Ching Truong, Trieu-Kien Lee, Hung-Peng Chang, Hsin-Chiu I Shou Univ Dept Informat Engn Kaohsiung Cty 840 Taiwan

In this paper, an algebraic decoding algorithm is proposed to correct all patterns of four or fewer errors in the binary (41, 21, 9) quadratic residue (QR) code. The technique needed here to decode the (41, 21, 9) QR code is different from the algorithms developed in [I.S. Reed, T.K. Truong, X. Chen, X. Yin, The algebraic decoding of the (41, 21, 9) quadratic residue code, IEEE Transactions on Information Theory 38 (1992) 974-986]. This proposed algorithm does not require to solve certain quadratic, cubic, and quartic equations and does not need to use any memory to store the five large tables of the fundamental parameters in GF(2(20)) to decode this QR code. By the modification of the technique developed in [R. He, I.S. Reed, T.K. Truong. X. Chen, Decoding the (47, 24. 11) quadratic residue code, IEEE Transactions on Information Theory 47 (2001) 1181-1186], one can express the unknown syndromes as functions of the known syndromes. With the appearance of known syndromes, one can solve Newton's identities to obtain the coefficients of the error-locator polynomials. Besides, the conditions for different number of errors of the received words will be derived. Computer simulations show that the proposed decoding algorithm requires about 22% less execution time than the syndrome decoding algorithm. Therefore, this proposed decoding scheme developed here is more efficient to implement and can shorten the decoding time. (C) 2009 Elsevier Inc. All rights reserved.

关键词： quadratic residue code Unknown syndrome Algebraic decoding Cyclic code Error-locator polynomial

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A cyclic weight algorithm of decoding the (47,24,11) quadratic residue code

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INFORMATION SCIENCES 2012年 197卷 215-222页

作者： Lin, Tsung-Ching Lee, Hung-Peng Chang, Hsin-Chiu Truong, Trieu-Kien Fortune Inst Technol Dept Informat Sci & Informat Engn Kaohsiung 831 Taiwan I Shou Univ Dept Informat Engn Kaohsiung 840 Taiwan

In this article, a novel table lookup decoding algorithm, called the cyclic weight (CW) decoding algorithm, is developed to facilitate faster decoding of the binary systematic (47,24,11) quadratic residue (QR) code. It is based on the property of cyclic codes together with the weight of syndromes. This new algorithm requires a lookup table which consists of 20.43 Kbytes. The advantage of the CW decoding algorithm over the previous table lookup method is that the memory size of the proposed lookup table is only about 1.89% of the lookup table needed in the decoding algorithm of Chen's et al. These facts lead to significantly reduce the decoding complexity in terms of CPU time while maintaining the capability to correct up to five errors. Simulation results show that the decoding speed of the proposed algorithm is much faster than that of the algorithm of Chen et al. (C) 2012 Elsevier Inc. All rights reserved.

关键词： quadratic residue code Compact lookup table Cyclic weight algorithm Syndrome decoder Error pattern

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THE ALGEBRAIC DECODING OF THE (41, 21, 9) quadratic residue code

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IEEE TRANSACTIONS ON INFORMATION THEORY 1992年第3期38卷 974-986页

作者： REED, IS TRUONG, TK CHEN, XM RAND CORP SANTA MONICACA 90406 MITRE CORP BEDFORDMA 01730 CALTECH JET PROP LABCOMMUN SYST RES SECTPASADENACA 91109

A new algebraic approach for decoding the quadratic residue (QR) codes, in particular the (41, 21, 9) QR code is presented. The key ideas behind this decoding technique are a systematic application of the Sylvester resultant method to the Newton identities associated with the code syndromes to find the error-locator polynomial, and next a method for determining error locations by solving certain quadratic, cubic and quartic equations over GF(2m) in a new way which uses Zech's logarithms for the arithmetic. The algorithms developed here for Zech's logarithms save a substantial amount of computer memory by storing only a table of the fundamental parameters instead of a complete table of Zech's logarithms. These algorithms are suitable for implementation in a programmable microprocessor or special-purpose VLSI chip. It is expected that the algebraic methods developed here can apply generally to other codes such as the BCH and Reed-Solomon codes.

关键词： ERROR-CORRECTION CODING THEORY CYCLIC code quadratic residue code FINITE FIELD

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Algebraic decoding of the (71, 36, 11) quadratic residue code

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IET COMMUNICATIONS 2016年第6期10卷 734-738页

作者： Lin, Tsung-Ching Chang, Hsin-Chiu Li, Yong Chang, Jack Trieu-Kien Truong I Shou Univ Dept Informat Engn Kaohsiung 82001 Taiwan Chongqing Univ Posts & Telecommun Chongqing Key Lab Mobile Commun Technol Chongqing 400065 Peoples R China Intellectual Property Grp Baker Hosterler WA USA

In this study, a new approach is developed to facilitate faster decoding of a binary systematic (71, 36, 11) quadratic residue (QR) code. In this decoder, it simplifies the step of calculating the condition and avoids calculating the unknown syndrome, thereby yielding a fast algebraic decoder for correcting four possible errors. Moreover, while using the proposed algorithm, if uses the channel measurement information proposed by Chase to sequentially invert the bits of the received word until one of the errors is cancelled for the five-error case and apply the new algebraic decoding algorithm mentioned above to correct the remaining four errors, the algorithm has been verified through a software simulation in C-language. The simulation shows that the decoding scheme developed here is more efficient than the previous decoding algorithm developed for the (71, 36, 11) QR code and it is naturally suitable for software implementation.

关键词： algebraic codes algebraic decoding algorithm quadratic residue code software simulation binary systematic quadratic residue code decoding channel measurement information C-language BCH codes codes

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Fast Algebraic Decoding of the (89,45,17) quadratic residue code

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IEEE COMMUNICATIONS LETTERS 2011年第2期15卷 226-228页

作者： Lin, Tsung-Ching Su, Wen-Ku Shih, Pei-Yu Truong, Trieu-Kien I Shou Univ Dept Informat Engn Kaohsiung Country 84001 Taiwan

In this letter, the algebraic decoding algorithm of the (89, 45, 17) binary quadratic residue (QR) code proposed by Truong et al. is modified by using the efficient determination algorithm of the primary unknown syndromes. The correctness of the proposed decoding algorithm is verified by computer simulations and the use of two corollaries. Also, simulation results show that the CPU time of this algorithm is approximately 4 times faster than that of the previously mentioned decoding algorithm at least. Therefore, such a fast decoding algorithm can now be applied to achieve efficiently the reliability-based decoding for the (89, 45, 17) QR code. Finally, the performance of its algebraic soft-decision decoder expressed in terms of the bit-error probability versus E(b)/N(o) is given but not available in the literature.

关键词： Berlekamp-Massey algorithm quadratic residue code unknown syndrome error-locator polynomial

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On Decoding of the (89, 45, 17) quadratic residue code

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IEEE TRANSACTIONS ON COMMUNICATIONS 2013年第3期61卷 832-841页

作者： Wang, Lin Li, Yong Truong, Trieu-Kien Lin, Tsung-Ching Xiamen Univ Dept Commun Engn Xiamen 361005 Peoples R China Chongqing Univ Posts & Telecommunicat CQUPT Chongqing Peoples R China I Shou Univ Dept Informat Engn Kaohsiung 84001 Taiwan Natl Sun Yat Sen Univ Dept Comp Sci & Engn Taipei Taiwan

In this paper, Three decoding methods of the (89, 45, 17) binary quadratic residue (QR) code to be presented are hard, soft and linear programming decoding algorithms. Firstly, a new hybrid algebraic decoding algorithm for the (89, 45, 17) QR code is proposed. It uses the Laplace formula to obtain the primary unknown syndromes, as done in Lin et al.'s algorithm when the number of errors v is less than or equal to 5, whereas Gaussian elimination is adopted to compute the unknown syndromes when v >= 6. Secondly, an appropriate modification to the algorithm developed by Chase is also given in this paper. Therefore, combining the proposed algebraic decoding algorithm with the modified Chase-II algorithm, called a new soft-decision decoding algorithm, becomes a complete soft decoding of QR codes. Thirdly, in order to further improve the error-correcting performance of the code, linear programming (LP) is utilized to decode the (89, 45, 17) QR code. Simulation results show that the proposed algebraic decoding algorithm reduces the decoding time when compared with Lin et al.'s hard decoding algorithm, and thus significantly reduces the decoding complexity of soft decoding while maintaining the same bit error rate (BER) performance. Moreover, the LP-based decoding improves the error-rate performance almost without increasing the decoding complexity, when compared with the new softdecision decoding algorithm. It provides a coding gain of 0.2 dB at BER = 2 x 10(-6).

关键词： Berlekamp-Massey algorithm Gaussian elimination quadratic residue code Chase algorithm linear programming

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Decoding the (31,16,7) quadratic residue code in GF(2∧5)

Decoding the (31,16,7) Quadratic Residue Code in GF(2∧5)

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4th International Conference on Computer Science and Education

作者： Lin, Tsung-Ching Chu, Shao-I Chang, Hsin-Chiu Lee, Hung-Peng I Shou Univ Dept Informat Engn Kaohsiung Kaohsiung Cty Taiwan

ISBN: (纸本)9781424435197

The binary QR codes are well known for their good behavior. The proposed algebraic decoding algorithm for decoding the (31, 16, 7) QR code with reducible generator polynomial is able to correct up to three errors in the finite field GF(2(5)). The proposed algorithm is based on an application of the decoding algorithm given by Truong et al. and Chen et al. to modify the decoding algorithm proposed by Reed el al. All syndromes in the error-locator polynomial are computed in the finite field GF(2(5)). Thus, the decoding time can be reduced. Moreover, the simulation results for comparing the proposed decoding algorithm with decoding algorithm given by Reed et al. are given. This algorithm is suitable for implementation in a programmable microprocessor or special-purpose VLSI chip.

关键词： quadratic residue code unknown syndrome decoding algorithm cyclic code error pattern

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Using FPGA-Based quadratic residue code to Improve the Reliable of Wireless Robot Control System 32

Using FPGA-Based Quadratic Residue Code to Improve the Relia...

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32nd Chinese Control And Decision Conference (CCDC)

作者： Xie, Shanggang Huang, Jiye Liu, Peng Chang, Hsin-chiu Hangzhou Dianzi Univ Dept Elect & Informat Hangzhou 310018 Peoples R China Hangzhou Dianzi Univ Key Lab RF Circuits & Syst Hangzhou 310018 Peoples R China

ISBN: (纸本)9781728158556

In the field of wireless mobile robots, the control code of the robot needs to be transmitted through a reliable wireless channel with low latency, strong interference resistance, simple handshake, and short data length. Thus, we adopt the (71, 36, 11) quadratic residue (QR) error correction code to encode the control code of wireless mobile robots to improve the reliability of wireless mobile robot control system. Among them, we analyze the software encoding and decoding algorithms of (71, 36, 11) QR code, then we perform hardware-oriented optimization and give a specific FPGA-based decoding framework design. Then we implement the proposed architecture on Intel Arria10 10AX115U4F45I1SG FPGA, after this we calculate the decoding time of the proposed framework and compare it with software decoding, this acceleration framework is 2275 times faster than software decoding based on Intel i5-6500 processor. At the same time, after adopting this scheme, it will be beneficial to the power consumption and communication distance of wireless mobile robots. Meanwhile, the FPGA parallel acceleration design ensures low transmission latency.

关键词： Wireless Mobile Robot FPGA Acceleration quadratic residue code

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Fast Decoding of the (41,21,9) quadratic residue code Without Computing Unknown Syndromes 10

Fast Decoding of the (41,21,9) Quadratic Residue Code Withou...

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10th International Conference on Wireless Communications and Signal Processing (WCSP)

作者： Luo, Chunlan Wu, Yi Chang, Hsin-chiu Yang, Zheng Shen, Lianfeng Xing, Song Fujian Normal Univ Fujian Prov Engn Technol Res Ctr Photoelect Sensi Fuzhou Fujian Peoples R China Southeast Univ Natl Mobile Commun Res Lab Nanjing Jiangsu Peoples R China Calif State Univ Los Angeles Dept Informat Syst Los Angeles CA 90032 USA

ISBN: (纸本)9781538661192

In this paper, an improved hard-decision scheme is proposed to facilitate a faster decoding of the binary (41, 21, 9) quadratic residue (QR) code. Based on eliminating the unknown syndromes in Newton's identities, the proposed algorithm can directly calculate the coefficients of the error-locator polynomials when there are 2, 3 or 4 errors in the received word. Additionally, compared to the conventional works, the simplification of the occurrence conditions for 1, 2 or 3 errors in the received word is also presented in our proposed algorithm, which can further reduce the decoding time. Simulation results show that the proposed algorithm can achieve a significant reduction in computational complexity in comparison with the conventional algebraic decoding algorithms (ADAs) like Lin's algorithm in [8], while maintaining the good error performance.

关键词： quadratic residue code algebraic decoding unknown syndrome Newtonian identity error-locator polynomial

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Algebraic and linear programming decoding of the (73,37,13) quadratic residue code

Algebraic and linear programming decoding of the (73,37,13) ...

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IEEE International Conference on Communications (ICC)

作者： Li, Yong Liu, Hongqing Chen, Qianbin Truong, Trieu-Kien Chongqing Inst Posts & Telecommun Chongqing Key Lab Mobile Commun Technol Chongqing 400065 Peoples R China I Shou Univ Dept Informat Engn Kaohsiung Taiwan

ISBN: (纸本)9781479920037

In this paper(1), a method to search the subsets I and J needed in computing the unknown syndromes for the (73, 37, 13) quadratic residue (QR) code is proposed. According to the resulting I and J, one computes the unknown syndromes, and thus finds the corresponding error-locator polynomial by using an inverse-free BM algorithm. Based on the modified Chase-II algorithm, the performance of soft-decision decoding for the (73, 37, 13) QR code is given. This result is never seen in the literature, to our knowledge. Moreover, the error-rate performance of linear programming (LP) decoding for the (73, 37, 13) QR code is also investigated, and LP-based decoding is shown to be significantly superior in performance to the algebraic soft-decision decoding while requiring almost the same computational complexity.

关键词： Berlekamp-Massey algorithm quadratic residue code Chase algorithm linear programming

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